4013
The Journal of Experimental Biology 204, 4013–4022 (2001)
Printed in Great Britain © The Company of Biologists Limited 2001
JEB3531
Review
Modelling the biomechanics and control of sphincters
Marcel Heldoorn1,*, Johan L. Van Leeuwen2 and Jan Vanderschoot3
1Department of Neurosurgery, Leiden University Medical Center (LUMC), Wassenaarseweg 62, PO Box 9604,
NL-2300 RC Leiden, The Netherlands, 2Experimental Zoology Group, Wageningen Institute of Animal Sciences
(WIAS), Wageningen University, Marijkeweg 40, PO Box 338, NL-6700 AH Wageningen, The Netherlands and
3Department of Medical Informatics, Leiden University Medical Center (LUMC), Wassenaarseweg 62,
PO Box 2086, NL-2301 CB Leiden, The Netherlands
*e-mail: [email protected]
Accepted 19 September 2001
Summary
This paper reviews current mathematical models of
sphincters and compares them with a new spatial
neuromuscular control model based on known
physiological properties. Almost all the sphincter models
reviewed were constructed as a component of a more
extensive model designed to mirror the overall behaviour
of a larger system such as the lower urinary tract. This
implied less detailed modelling of the sphincter
component. It is concluded that current sphincter models
are not suitable for mimicking detailed interactions
between a neural controller and a sphincter. We therefore
outline a new integrated model of the biomechanics and
neural control of a sphincter. The muscle is represented
as a lumped-mass model, providing the possibility of
applying two- or three-dimensional modelling strategies.
The neural network is a multi-compartment model that
provides neural control signals at the level of action
potentials.
The integrated model was used to simulate a uniformly
activated sphincter and a partially deficient innervation
of the sphincter, resulting in a non-uniformly activated
sphincter muscle. During the simulation, the pressure
in the sphincter lumen was prescribed to increase
sinusoidally to a value of 60 kPa. In the uniformly
activated situation, the sphincter muscle remains closed,
whereas the partially denervated sphincter is stretched
open, although the muscle is intact.
Key words: sphincter, muscle, neuromuscular control, biomechanics,
quantitative modelling, neural network, lumped-mass model, multicompartment model.
Introduction
The main mechanical functions of sphincter muscles are
controlled constrictions and relaxations. Sphincters are used
for temporary mechanical separation of compartments ranging
from a valve-like function in one-directional transport of food
in the gastrointestinal tract to a water-tight seal of the urinary
bladder or the rectum. Different functions and topologies
demand architectural adaptations of a sphincter and
adaptations of its neural control.
Mathematical models of sphincters can be used to study
various aspects of these adaptations. In most current models,
the sphincter is implicitly included in a larger system,
for instance in a model of the lower urinary tract
(Bastiaanssen et al., 1996a; Bastiaanssen et al., 1996b). In
these systems, the model of the sphincter is a simple
structure, because the complete models were designed to
simulate the collective behaviour of a larger system and were
not intended to study the detailed internal dynamics of a
sphincter. Most published models of sphincters include only
one ‘muscle unit’. The muscle units range from a resistance
with an on/off switch control to a resistance with muscle-like
dynamics and most have a one-dimensional geometry. To
study detailed aspects of a distributed parameter system, such
as a sphincter or a neural network, more complex spatial
models of several interacting units are needed. We found only
one spatial model of a sphincter in the literature (Gielen,
1998).
After reviewing current models of sphincter, we outline a
new integrated neuromuscular model of a sphincter and its
neural control that allows detailed study of various aspects of
this system, such as the spatial distribution of different muscle
fibre types and stretch receptors, the effects of spatial
neuromuscular defects, neural activation and dynamics. This
integrated neuromuscular model consists of two parts. The
muscle is represented as a lumped-mass model offering the
possibility of applying two- or three-dimensional modelling
strategies. The neural network is a multi-compartment model
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M. Heldoorn, J. L. Van Leeuwen and J. Vanderschoot
that examines signal flow in the neural controller at the level
of action potentials.
Comparison of models in literature
Sphincter muscle models
Current sphincter models are compared in Table 1. All
current models of sphincters are portions of models designed
to represent the lower urinary tract, except for a model of the
pupil of the eye (Usui and Hirata, 1995) and the finite-element
model of Gielen (1998). In many models of the lower urinary
tract, the sphincter is not modelled separately but is integrated
into the urethra (Drolet and Kunov, 1975; Fröhlich et al., 1977;
Hosein and Griffiths, 1990; Valentini et al., 1992; Van Duyl,
1993; Hübener and Van Mastrigt, 1994). In these models,
closing and opening of the urethra are realised by multiplying
the outflow by the value of a simple block function (with
possible function values of 0 or 1) (Fröhlich et al., 1977; Van
Duyl, 1993) or by applying a simple pressure threshold that,
when exceeded, allows urine outflow (Drolet and Kunov, 1975;
Hosein and Griffiths, 1990; Valentini et al., 1992; Hübener and
Van Mastrigt, 1994).
In some models (Hosein and Griffiths, 1990; Van Duyl,
1993; Hübener and Van Mastrigt, 1994), the mechanical
behaviour of the urethra is represented by a non-linear
relationship between bladder pressure and flow rate, known as
the urethral resistance relationship (Griffiths, 1980).
In three models (Usui and Hirata, 1995; Bastiaanssen et al.,
1996b; Van Duin et al., 1998; Van Duin et al., 1999), a
B
A
C
EE
EE
CE
EE
CE
VE
EE
CE
Fig. 1. Diagram of the muscle units used in sphincter models: (A) as
in Van Duin et al. (1998, 1999); (B) as in Bastiaanssen et al.
(1996a,b); (C) as in Usui and Hirata (1995). CE, contractile element;
EE, elastic element; VE, viscous element. Arrows show directions of
external forces on the systems.
different approach was followed. A sphincter was modelled
using visco-elastic elements and activity-dependent contractile
elements (with included visco-elasticity). These models show
muscle-like dynamics and will be discussed in the next section.
Gielen (1998) proposed a spatial finite-element model for
skeletal muscles that was also used to represent a sphincter.
Single-unit models with muscle dynamics
Hill’s three-element visco-elastic model is a commonly used
representation of muscle. The mechanical behaviour of the
muscle is described by a Hill-type contractile element (CE)
based on Hill’s characteristic equation (Hill, 1938), which was
later shown to be a phenomenological description of the
Table 1. Comparison of sphincter models
Muscle model
Sphincter sub-model
On/off flow switch
Pressure threshold
Urethral resistance relationship
Visco-elastic elements
Mass inertia
Finite element method
Multi-unit muscle
Neural model
Neural control
Open-loop control
Closed-loop control
Multi-unit neural control
A
B
•
•
•
C
•
•
•
•
D
E
•
•
•
•
•
•
F
•
•
•
•
G*
H
I
J
K
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
A, Drolet and Kunov, 1975; B, Fröhlich et al., 1977; C, Hosein and Griffiths, 1990; D, Valentini et al., 1992; E, Van Duyl, 1993; F, Hübener
and Van Mastrigt, 1994; G, Usui and Hirata, 1995; H, Bastiaanssen et al., 1996a; Bastiaanssen et al., 1996b; I, Van Duin et al., 1998; Van Duin
et al., 1999; J, Gielen, 1998; K, this article.
Sphincter sub-model, sphincter not integrated in the urethra, but a separate sub-model of a sphincter present; on/off flow switch, flow was
controlled by an on/off switch; pressure threshold, flow starts when exceeding the pressure threshold; viscoelastic elements, Hill-type muscle
dynamics; mass inertia, utilising Newton’s second law to calculate the accelerations of the characteristic masses; finite-element method, the
finite-element method was used to solve the continuum model numerically; neural control, neural control included; open-loop control, neural
signals were prescribed; closed-loop control, multi-centre neural control with mutual influences and feedback.
*This model represents the pupil of the eye, consisting of a dilator and a sphincter muscle.
Modelling the biomechanics and control of sphincters
behaviour of the actin and myosin proteins (Huxley and
Niedergerke, 1954; Gordon et al., 1966). It includes elastic
elements (EEs) (see Fig. 1A). If the CE is not activated, the
EE represents the passive elastic properties of the muscle.
Knowing the strain rate of the elastic elements and active
state, the mechanical stress in the muscle can be derived (see
Fig. 1A). This stress is assumed to be uniform in the muscle.
The normal stress on the luminal fluid generated by the
sphincter muscle can be calculated when its state of activation,
radius and width are known. The normal stress is assumed to
be in force equilibrium with the stress on the muscle as a result
of the intraurethral pressure (Van Duin et al., 1998; Van Duin
et al., 1999), which implies that inertial effects are ignored.
This type of model is therefore known as an equilibrium
model. Bastiaanssen et al. (1996a,b) also used an equilibrium
model, with one EE (see Fig. 1B). The force production of the
CE in the sphincter depends on the length of the CE, the
specific shortening velocity and the state of activation of the
CE. The passive EE and the active cross-bridges contribute,
in parallel, to the tensile stress in the sphincter. A
homogeneous distribution and shortening velocity of the CE
and the EE in the muscular layer of the sphincter is assumed.
In addition, a uniform activation of the CE is assumed. As a
result, the tensile stress in the sphincter can be described by
the stress production of a single muscle fibre. The tensile
stress of the single muscle fibre equals the sum of the active
tensile stress due to the cross-bridges in the muscle fibres and
the passive tensile stress of connective tissue and epithelium
due to the EE.
Usui and Hirata (1995) included a viscous element (VE) in
the muscle unit of their model pupil (see Fig. 1C). The
sphincter and dilator models (the pupil includes a sphincter and
a dilator muscle with radially directed muscle fibres) were both
constructed using the same basic units (see Fig. 1C), but using
different parameters, and were combined according to the
structure of the pupillary muscle. The pupillary muscle system
was simplified to a one-dimensional push/pull structure using
Newton’s second law to calculate the variation in pupil radius
from the forces exerted by the sphincter and dilator muscles on
a characteristic mass.
Multi-unit model with muscle dynamics
Gielen (1998) proposed a continuum description for a
contracting skeletal muscle in which the distributed moments
model of Zahalak (1981) was implemented to derive the
constitutive equations. In this approach, the cross-bridge forces
are essentially described using the Huxley (1957) description.
The force of a half-sarcomere is derived by summation of the
contribution of the available cross-bridges. The continuum
description was solved numerically using the finite-element
method. Mass inertia was ignored in this model. This approach
is promising, but a test of the model (Gielen, 1998) with a
prescribed axi-symmetric load and an initial axi-symmetric
sphincter geometry was only partially successful because
significant deviations were found from the ideal axi-symmetric
solution. The deviations were attributed to limitations in the
4015
applied mesh and the associated choice of the properties of the
elements.
Neural control of the model sphincter
The neural control systems of current sphincter models are
compared in Table 1: some models lack a neural controller for
the sphincter (Drolet and Kunov, 1975; Fröhlich et al., 1977;
Van Duyl, 1993; Gielen, 1998), and the process of opening and
closing depends only on an on/off switch that, in some models,
was coupled to a threshold pressure (Fröhlich et al., 1977; Van
Duyl, 1993). Valentini et al. (1992) and Hübener and Van
Mastrigt (1994) prescribed the neural excitation of the urethra
as a function of time. This approach is known as an open-loop
control system.
Closed-loop neural control
Hosein and Griffiths (1990) postulated a neural control
system involving two mutually inhibitory control regions,
leading to a bi-stable (storage-voiding) system. The reflexes
were driven by a single afferent signal with contributions from
bladder wall tension and urethral distension.
Usui and Hirata (1995) modelled the portion of the
autonomic nervous system innervating the pupil: the
parasympathetic and sympathetic input to the sphincter and
the dilator muscle, respectively. The parasympathetic and
sympathetic nervous activities were mutually inhibitory.
Bastiaanssen et al. (1996a) used a qualitative model of the
neuronal circuitry involved in the control of the uropoietic
system (Kinder et al., 1995) to build a detailed quantitative
model of the different anatomical structures involved in the
system.
Van Duin et al. (1998, 1999) presented a less complicated
alternative description of the neural control of the lower
urinary tract. The input parameters to the neural model were
related to the length of the bladder and urethral wall, to the
pressure in the bladder and to the stress in the bladder wall.
Current neuromuscular sphincter models
All previous models were designed to mirror the overall
behaviour of a large system. This implied less-detailed
modelling of components so that, for instance, the pressure in
the sphincter is reflected by only one value. For some purposes,
this coarse modelling is appropriate. However, the pressure in
a sphincter muscle follows a gradient, rather than having a
single value, and the state of activation varies over the
sphincter (Podnar and Vodus̆ek, 1999; Podnar et al., 1999;
Podnar and Vodus̆ek, 2000). In the literature, the term
distributed parameter system is often used for this type of
system. Distributed parameter systems can be modelled by a
continuum approach (e.g. Gielen, 1998). Alternatively, spatially
discrete (lumped) models can be used in which continuously
distributed parameters (properties) are concentrated in the
crossings of a spatial grid (spatial discretisation). For both
approaches, more details can be revealed by using a finer
mesh or grid at the expense of higher computer memory
requirements and a longer computation time. By expressing the
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M. Heldoorn, J. L. Van Leeuwen and J. Vanderschoot
Fig. 2. Diagram of the model sphincter: a twoC
A
B
dimensional system of point masses. (A) The
masses were assumed to be interconnected by
g
a d
tensile elements (with zero cross-sectional area)
h
b e
thought to represent the circularly arranged muscle
j
c f
fibre bundles and the radially oriented connective
tissue. (B) Detail of the shaded portion in A. The
masses were connected to either four (corner
ed
f
point) or two (mid point) tensile elements, with the
ab
c
exception of the corner point masses on the inner
or outer border, which are connected to only three
g hj
tensile elements. The tensile elements were
assumed to have a constant curvature along their
length, as determined by the locations of two adjacent corner points and one intermediate mid point. Fluid elements were enclosed by the
tensile elements through four corner points and four mid points. The pressure was assumed to be constant in any one element. The motion of
the masses was calculated by application of Newton’s second law. The acceleration of each mass was calculated from the forces in the
connecting tensile elements (red arrows) and the forces due to the pressure in the adjacent fluid compartments (blue arrows). (C) Mapping of
the neurons in the neural network to the muscle elements in the sphincter. The neurons in the neural network (labelled a–j in the upper square)
correspond to the muscle elements in the sphincter. The vertical columns of neurons innervate the circular layers of the muscle, starting from
the outside, whereas the horizontal rows innervate the radial segments of the muscle, starting at the 09:00 h position and proceeding in a
clockwise direction.
state of a sphincter in a single value, much information is
ignored. To study the finely distributed properties of a typical
distributed parameter system, less coarse models are needed.
Current sphincter models are not suitable for mimicking the
detailed interactions between neural control and sphincter.
In the next section, we propose a spatial model for the
neuromuscular control of sphincters in which a multicompartmental model of a neural network is connected to a
lumped-mass model of a sphincter muscle.
The lumped-mass sphincter model
The sphincter is modelled as a two-dimensional system of n
point masses. The masses are assumed to be interconnected by
tensile elements (with zero cross-sectional area), thought to
represent muscle fibre bundles and connective tissue. The topology
of masses and tensile elements is shown in Fig. 2A. The masses
are connected to either four (corner point) or two (mid point) tensile
elements, with the exception of the corner point masses on the inner
or outer border, which are connected to only three tensile elements.
The tensile elements are assumed to have a constant curvature
along their length, as determined by the locations of two adjacent
corner points and one intermediate mid point. Fluid elements are
enclosed by the tensile elements through four corner points and
four mid points. The fluid is assumed to be slightly compressible,
described by a bulk modulus. The pressure is assumed to be
constant in any one element. Both the active and the passive elastic
elements have non-linear properties. For the active component of
muscle tissue, non-linear force/velocity (Hill-type) relationships
have been assumed, whereas length-dependence on filament
overlap (and thus force development) is included. The stress of the
passive elements is non-linearly related to the strain. Tensile
elements are assumed to be arranged in a circular fashion. Muscle
fibres may extend over one or more tensile elements. Their tensile
stresses are assumed to depend on passive tensile properties,
passive viscous properties and activity-related dependencies of
longitudinal strain and strain rate, described after Van Leeuwen
and Kier (1997). The activity of the tensile elements is described
by first-order dynamics with the local axonal firing rate as input
(see Bastiaanssen, 1996). The radial tensile elements are described
by passive elastic and viscous properties only. The motion of the
masses is calculated by application of Newton’s second law. The
acceleration of each mass is calculated from the forces in the
connecting tensile elements and the forces due to the pressure in
the adjacent fluid compartments. Briefly, the set of n coupled
second-order differential equations is first rewritten to an
equivalent system of 2n first-order differential equations and then
numerically solved using the Adams–Bashford–Moulton method
(Burden et al., 1981; Mathews, 1992).
Neural network model
Compartmental modelling
When constructing detailed neural models that explicitly
include many of the potential complexities of a cell, the usual
approach is to divide the neuron into a finite number of
interconnected compartments. Each compartment is modelled
with equations describing an equivalent electrical circuit (Rall,
1959). With the appropriate differential equations for each
compartment, depending on its active and passive electrical
properties, one can model the behaviour of each compartment
and its interactions with neighbouring compartments.
Conceptually, electrically short segments are assumed to
be isopotential and are lumped into a single membrane
compartment with a specified resistance and capacitance, either
passive or active. Compartments are connected to one another
via a longitudinal resistivity according to the topology of the
tree describing the detailed morphology of the original neuron.
Hence, differences in physical properties (e.g. diameter,
membrane properties, etc.) and differences in potential occur
Modelling the biomechanics and control of sphincters
4017
Table 2. Ionic currents in the segments of the model motor neuron
Segment
Current
Characteristic equation
Reversal
potential, Eion
(mV)
Maximum
conductance, ḡion
(mS cm−2)
Soma segment
INa(S) = ḡNa(S)m3h (Vm–ENa(S))
IK(S) = ḡK(S)n4(Vm−EK(S))
IKFR(S) = ḡKFR(S)p2(Vm–EKFR(S))
115.0
−10.0
−10.0
140.0
35.0
35.0
Axonal segment
INa(IS) = ḡNa(IS)m3h(Vm–ENa(IS))
IK(IS) = ḡK(IS)n4(Vm–EK(IS))
115.0
−10.0
500.0
100.0
Dendritic segment
Ileak = ḡleak(Vm–Eleak)
10.613
0.3
Characteristics of the ion current definitions in the segments of the neuron model.
Soma segment currents: INa(S) (sodium), IK(S) (potassium), IKFR(S) (slow potassium). Axonal segment currents: INa(IS) (sodium), IK(IS)
(potassium).
Eion and ḡion, reversal potential and maximal conductance, respectively, of the Na+ (Na), K+ (K) and slow K+ (KFR) currents; Vm, membrane
potential. Active currents are not present in the dendritic segment.
The total ion current for the motor neuron model in the soma is: Iion=gNa(S)(Vm–ENa(S))+gK(S)(Vm–EK(S))+gKFR(S))(Vm–EKFR(S)), which
equals ḡNa(S)m3h(Vm–ENa(S))+ḡK(S)n4(Vm–EK(S))+ḡKFR(S)p2(Vm–EKFR(S)), where ḡion is the maximum conductance for the Na+ (Na), K+ (K) and
slow K+ (KFR) ion channels; m, h, n and p are time- and voltage-dependent parameters that determine the magnitude and time course of the
activation and inactivation of the respective currents.
The time- and voltage-dependence is given by: dη/dt=αη(1-η)-βηη=η∞-η/τη, where τη=1/αη+βη and t is time with η=m, h, n or p. The
magnitudes and time courses of the activation/inactivation parameters αη and βη are given in Table 3.
between compartments rather than within them. The length of
a compartment must be small enough for its electrotonic length
to be greater than 0.2λ (Segev et al., 1985) to consider it as
isopotential. The space constant λ is the length of a cable with
the same diameter (as the compartment) that has a membrane
resistance equal to the cytoplasmic resistance. Often, this
requires simulations with very large numbers of compartments.
A GENESIS (GEneral NEural SImulation System) (Wilson et
al., 1989; Bhalla and Bower, 1992; Bower and Beeman, 1994)
model of a cerebellar Purkinje cell was recently published that
uses 4550 compartments and 8021 channels (De Schutter and
Bower, 1994a; De Schutter and Bower, 1994b).
The computation time for a simulation depends on several
factors: the integration method, the precision of the method, the
size of the model and the total number of conductances. One
can simplify the model of a neuron to keep the number of
compartments (and the computation time) reasonable. The
classical way to simplify dendritic trees is to lump branches
together into an equivalent cylinder, i.e. to use one large dendrite
instead of several small ones. An extreme of this method is the
‘ball-and-stick’ model used in linear cable models: one spherical
somatic compartment and a large cylindrical compartment
equivalent to the axon, the dendrites, etc. This ‘ball-and-stick’
method was used to construct the neuron model in the following
section. The simplification of the motor neurons was motivated
by the fact that a more complicated model would make no
difference in the current simulation. In this simulation
experiment, the motor neurons are responsible only for the
uniform and synchronous activation of the sphincter muscle.
Neural structure
The model neuron consists of a somatic segment, a dendritic
segment and one axonal compartment. The dimensions of the
soma compartment were based on reconstructions of external
urethral sphincter motor neurons from Onuf’s nucleus (Sasaki,
1994). In the present model, the axon is represented by a sphere
with a diameter of 28.40 µm. Axonal segment length and
diameter are 100 µm and 5 µm, respectively. The length and
diameter of the dendritic segment are 100 µm and 4 µm,
respectively.
Passive and active parameters
The passive parameters of the neurons were set according to
the values reported by Fleshman et al. (1988) for cat α-motor
neurons. Specific membrane capacitance Cm was 1.0 µF cm–2,
and specific cytoplasmic resistance Ri was 70 Ω cm. The
neuron had values for the specific membrane resistance Rm of
11 kΩ cm2 for the dendrite and 0.225 kΩ cm2 for the other
compartments (soma and axonal segment).
Active properties were associated with the initial segment
and soma compartment only and consisted of five voltagedependent channel types modelled using Hodgkin–Huxley
equations (Hodgkin and Huxley, 1952). The conductances in
the neuron segments are given in Table 2. The axonal segment
contained a fast Na+ conductance and a fast delayed rectifier
(Barrett and Crill, 1980). The soma compartment contained a
fast Na+ conductance, a fast K+ conductance and a slow K+
conductance. The K+ conductances are analogous to those
described by Barrett et al. (1980) for cat motor neurons. Jones
and Bawa (1997) used the same composition for a comparison
between cat and human motor neurons. They suggested that
the biophysical properties governing rhythmic firing in human
motor neurons are similar to those of the cat (see Table 2 and
Table 3 for a detailed description of parameters).
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M. Heldoorn, J. L. Van Leeuwen and J. Vanderschoot
Table 3. Model parameters of ion channels in neuron compartments
Parameters characterising rate factor dynamics
Soma segment currents
Current
Rate factor
x1
x2
INa(S)
αm
βm
αn
βh
αn
βn
αp
βp
7
−18
0.15
40
0.4
0.16
3.5
0.035
−0.4
0.4
0
0
−0.02
0
0
0
αm
βm
αh
βh
αn
βn
4
−14
0.16
40
0.2
0.15
−0.4
0.4
0
0
−0.02
0
IK(S)
IKFR(S)
Axonal segment currents
INa(IS)
IK(IS)
x3
x4
x5
−1
−1
0
1
−1
−0.032
1
1
−17.5
−45
−34.26
−40
−20
−33.79
−55
50
−5
5
18.19
−10
−10
66.56
−4
−0.001
−1
−1
0
1
−1
−0.01
−10
−35
−37.78
−30
−10
−33.79
−5
5
18.14
−10
−10
71.86
The general form of the equation for the rate factors αη, βη=x1+(x2Vm)/x3+e(Vm+x4)/x5, where Vm is membrane potential.
The time- and voltage-dependence is given by: dη/dt=αη(1-η)–βηη=η∞−η/τη, where τη=1/αη+βη and t is time. In the above equations, η can
be substituted with the parameters m, h, n and p (see also Table 2).
The currents are defined in Table 2.
A postsynaptic conductance was implemented in the dendritic
segments, mimicking the time delay caused by transmitter release
and offering the possibility of simulating input from other
neurons. The model also offers the possibility of injecting current
in an arbitrary segment or neuron. The neuron model offers a
choice between two different interfaces between the neural
network and the muscle, a voltage-dependent presynaptic
transmitter release or a discharge-frequency-calculating interface.
Results
The primary aim of this simulation experiment was to
illustrate the advantages offered by this multi-unit model in
contrast to the limitations of previous single-unit sphincter
models presented in the literature. All simulations were carried
out using dedicated software in Borland Delphi Professional
version 4 on an Intel Pentium-II computer.
Description of the simulations
The integrated model was used to simulate (i) a normal
sphincter and (ii) a partially deficient innervation of the
sphincter. The normal situation was simulated with a fully
functional neural net providing a uniformly activated sphincter.
In the second simulation, one quarter of the sphincter was not
activated. Although this activation pattern is unlikely to occur
under physiological conditions, it was chosen to test the
robustness of the present model. In the experiments, the
pressure in the lumen of the sphincter was prescribed to
increase sinusoidally from 5 to 60 kPa. All neurons in the
network received a current input of 2.5 nA in pulses of 2 ms,
with an interspike interval of 55 ms, causing the neurons to fire
with a frequency of approximately 20 Hz.
The connection between some neurons and the
accompanying muscle elements was blocked to simulate a
partially denervated sphincter. Muscle properties did not
change between simulations. The results of the simulation are
shown in Fig. 3. Fig. 3A–D shows the results of the partial
activation experiment at various times (t=0, 350, 700 and
1050 ms), while Fig. 3E shows the state of the normally
activated model at 1050 ms.
The network activity is plotted on the top row in Fig. 3. Each
rectangle represents a neuron, forming a 3×7 matrix of
neurons. Each muscle element is innervated by one neuron.
The model sphincter consists of five circular layers. The three
outer layers represent muscle tissue, while the inside of the
sphincter is formed by two circular layers of passive tissue to
simulate the epithelium and the lamina propria (visco-elastic
connective tissue). The vertical columns of neurons innervate
the circular layers of the muscle, starting from the outside,
whereas the horizontal rows innervate the radial segments of
the muscle, starting at the 09:00 h position and proceeding in
a clockwise direction (see Fig. 2C). This configuration is
artificial but can easily be replaced if detailed anatomical data
become available. The two right-hand columns of the neural
network in Fig. 3A–D show a (prescribed) normalised activity
of zero in contrast to the network activity in the normal
situation (see Fig. 3E).
The network activity stimulated the sphincter, resulting in an
active state of the muscle elements (second row in Fig. 3). A
fully activated network resulted in a uniformly activated
sphincter (see Fig. 3E), whereas a partially inactivated network
resulted in a non-uniformly activated sphincter (see Fig. 3A–D).
The fibre strain (third row in Fig. 3) is low when t=0 ms.
The sphincter is pushed outwards by the increased luminal
Modelling the biomechanics and control of sphincters
A
B
C
D
E
0 ms
350 ms
700 ms
1050 ms
1050 ms normal
4019
Network activity
1
0.5
0
1
0.5
0
Active state
1
0.5
0
1
0.5
0
Fibre strain
0.4
0.2
0
0.03
0.02
0.01
0
Pressure (Pa)
10×104
5×104
0
10×104
5×104
0
Fibre stress (Pa)
4×105
2×105
0
5×105
0
Fig. 3. Simulation experiments with the neuromuscular sphincter model. (A–D) The results of the partial activation experiment at various
points in time (0, 350, 700 and 1050 ms, respectively); (E) the state of the uniformly activated model at 1050 ms as a control. The normalised
network activity is shown in the top row. The normalised active state, the fibre strain, the pressure and the fibre stress in the sphincter are
depicted in the subsequent rows. The colours in the graphs represent values indicated by the accompanying colour scales. In the pressure graph,
the circular lines represent the muscle fibres, while the radial lines represent passive connective tissue fibres. In the pressure panel, the lumen is
coloured, indicating the pressure in the lumen of the sphincter; the lumen has no value for strain, stress or active state.
pressure. This causes the strain in the sphincter wall to
increase. However, since a portion of the sphincter is activated
and tends to contract, only the non-activated portion of the
sphincter is stretched. This is clearly shown in Fig. 3D, in
which the strain in the activated portion is approximately zero,
but the stress is high. Thus, the strain graphs show that the nonactivated portion of the sphincter is stretched by the activated
contracting portion of the muscle elements and the increasing
pressure in the lumen. Counteracting forces act on the
sphincter, causing high pressure near the lumen and low
pressure on the outside of the muscle. The gradient is
proportional to the tensile stress (fifth row in Fig. 3) and is
inversely proportional to the radius of curvature of the layers
of muscle fibres. The smaller the curvature (at positions nearer
the middle point of the sphincter), the higher the pressure
gradient that can be generated.
A few remarks can be made on comparing the model states
at t=1050 ms in the experiment (Fig. 3D) and the control
(Fig. 3E). First, there is no difference in muscle properties;
both states reflect the effects of distinct activation patterns on
normally functioning sphincter tissue. Second, the strain in the
non-activated portion is so high that the non-activated portion
bulges out. By changing the uniformity of activation of the
muscle fibres, the sphincter loses its normal function.
In conclusion, when the muscle is intact and functioning, the
uniformly activated muscle is able to keep the sphincter closed,
although the internal luminal pressure is high. Conversely, the
partially denervated (non-uniformly activated) muscle is not
capable of remaining closed.
Discussion
The integrated neuromuscular model is divided into two
sub-models: a neural network and a sphincter muscle.
Neural network
We used a biophysically minimal motor neuron model to
simulate neuromuscular sphincter control. In future work, the
4020
M. Heldoorn, J. L. Van Leeuwen and J. Vanderschoot
motor neuron model needs to be extended since Feirabend et
al. (1997) have demonstrated the existence of gap junctions in
human Onuf’s nucleus, which is the main motor neuron
population innervating the urethral and anal sphincters. Gap
junctions are of particular importance in forming lowresistance pathways through which currents generated by
action potentials in one cell activate adjacent cells. The model
offers the possibility of adding gap junctions between
compartments of separate neurons. Although the physiological
experimental data of Sasaki (1991) were partially
implemented, the experimentally indicated anomalous inward
rectifying current in external urethral sphincter motor neurons
analogous to the hyperpolarization activated IQ t current
(Halliwell and Adams, 1982) (also known as Ih) (Mayer and
Westbrook, 1983) has not been incorporated.
The inferior olive nucleus shows similar characteristics: the
cells are coupled by electrotonic gap junctions located on the
dendrites (Llinás et al., 1974; Sotelo et al., 1974). Ih was also
found in the inferior olive nucleus. Detailed studies on Ih
have revealed its critical role in synchronised oscillations
(McCormick and Pape 1990; Soltesz et al., 1991; McCormick
and Huguenard, 1992; Bal and McCormick, 1997).
Consequently, future modelling needs to include these items.
Sensory feedback was implemented to represent a basic
reflex system. Nevertheless, during the simulations, the model
functions as an open-loop system because the input of the
strain receptor activity (=sensory feedback) into the network
was blocked to ensure uniform neural activity in the neural
network during the simulation experiment.
The simplification of the motor neurons was motivated (i)
by the minimal experimental data available about external
urethral sphincter motor neurons from Onuf’s nucleus and (ii)
by the fact that a more complicated model would make
no difference in the present simulation experiment. In this
simulation experiment, the motor neurons are responsible only
for the uniform and synchronous activation of the sphincter
muscle.
Sphincter muscle
The description of a multi-unit muscle as a two-dimensional
system of point masses has also been used in a simulation of
a non-circular muscle: the tentacle of squid (Van Leeuwen and
Kier, 1997). The simulation of the completely activated
sphincter model showed a circular muscle. Various references
in the literature indeed show circular (ultrasound) images of
the morphology of the (anal) sphincter (Law et al., 1990;
Tjandra et al., 1992; Gantke et al., 1993; Schäfer et al., 1994).
The model presented here is not limited to axi-symmetric
problems, although the initial geometry chosen for the
modelled structure is. The model is able to simulate a variety
of (sphincter) muscle forms, such as a U-shaped geometry. In
our model, muscle fibre activation is not mediated by
biophysical correlates such as presynaptic transmitter release,
postsynaptic ligand-activated conductance, end-plate potential
and Ca2+ action potential in T-tubules, but only by the
discharge frequency of the motor neuron: activation of a motor
neuron results in contraction of the fibres it innervates.
Relaxation or lengthening of muscle elements can be achieved
passively (by the lumen pressure), for which it is necessary to
reduce the motor neuron discharge (Berne and Levy, 1994).
The tension generated in a muscle motor unit will, therefore,
be a function of the discharge rate of its motor neuron.
The model presented here can handle more advanced
realistic sphincter morphologies. The subdivision in separate
circumferential muscular elements opens the perspective for
realistic representations of, for example, the distribution of
activation. Local variations in the sphincter morphology, such
as muscle fibre types (Light et al., 1997; Krier et al., 1988;
Oelrich, 1983; Schrøder and Reske Nielsen, 1983; Oelrich,
1980; Bowen et al., 1976), muscle and connective tissue fibre
arrangements and neural activation can be simulated using the
present model. This model cannot deal with muscle fibres
traversing the defined layers. Future extensions of the model
might support three-dimensional architectures.
The simulation
The experiments described in this paper were simulated
further than t=1050 ms, but no significant changes could be
detected in the remainder of the simulation. Test runs
confirmed that the present model is stable for at least 20 s.
Physical stability
In the simulation of uniform muscle activation, the models
evolve to an equilibrium configuration in which all muscle
fibres are shortened. The simulation of partial activation is an
extreme test of numerical and physical stability. Partial
activation leads to a physical state that is initially far from
equilibrium. Neighbouring fibres (in a circumferential arc)
display completely different behaviour. The activated fibres
contract, resulting in stretching of the inactive fibres. The
abrupt transition between the activated and the non-activated
portion results in a large deviation in axial symmetry and in
abrupt jumps in the local strain. The model provides a
numerically stable solution of the highly transient behaviour
towards the new equilibrium situation (with the applied steady
external load). We expect the model to behave equally well in
less extreme physical circumstances.
Future verification of the model
At present, detailed experimental data for the strain, stress
and pressure distribution in sphincter muscles are rare,
presumably because of experimental difficulties. Future
research requires advanced experimental techniques to collect
the data necessary to verify the results of the model. Promising
experimental techniques seem to be high-resolution nuclear
magnetic resonance tagging, ultrasound imaging, the
application of miniature ultrasound crystals for distance
measurements and distributed needle electromyography. The
limiting factor of current imaging research is the low
resolution; much higher resolution is needed for our purposes.
The present model forms the best available prediction of
sphincter deformations and may be a useful tool in the choice
Modelling the biomechanics and control of sphincters
of the experimental arrangement required to study the spatial
behaviour of sphincters. The need for detailed experimental
data could direct future experimental research in different areas
such as those mentioned above.
However, using advanced models in the absence of
experimental data is the second-best option for studying
complex non-linear systems with distributed properties. The
model presented here points to an interesting phenomenon:
partial activation of a sphincter muscle leads to work done on
the inactive portion by the activated portion and, consequently,
to a less effective sphincter muscle. This simulation illustrates
an application and the potency of integrated multi-unit models
in contrast to single-unit muscle models.
Discussion of the dynamic stability and the consequences of
non-linearity to the control of movement, although beyond the
scope of the present paper, is a logical next step. In future
research, this type of model supported by detailed experimental
data might be helpful in biological and/or clinical applications.
The authors are indebted to Professor Dr A. R. Bakker,
Professor Dr E. Marani and Professor Dr A. A. B. Lycklama à
Nijeholt for useful suggestions.
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